For questions about geometric shapes, congruences, similarities, transformations, as well as the properties of classes of figures, points, lines, angles.

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0answers
26 views

Intuitive argument for the transitivity of $PGL(n+1)$ acting on $\mathbb{P}^n$

In spherical geometry we can consider the action of $\operatorname{O}(n+1)$ on the unit sphere $\mathbb{S}^n$. It's easy to see that this action is transitive, because for any two points $x,y \in ...
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1answer
37 views

Evaluating a Trigonometric Expression involving Periodicity

Evaluate: $$\dfrac{\csc(90+\theta)+\cot(450+\theta)}{\csc(450-\theta)-\tan(180+\theta)}+\dfrac{\tan(180+\theta)+\sec(180-\theta)}{\tan(360-\theta)-\sec(-\theta)}$$ I simplified this into ...
2
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3answers
60 views

How to find the sum of distances so that it is minimal?

Question: $A$ and $B$ are two points on the same side of a line $l$. Denote the orthogonal projections of $A$ and $B$ onto $l$ by $A^\prime$ and $B^\prime$. Suppose that the following distance are ...
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1answer
18 views

Generalization of the Saccheri-Legendre Theorem Proof

So I'm working on generalizing the Saccheri-Legendre Theorem to convex $n$-gons. $\underline{\text{Saccheri-Legendre Theorem:}}$ The sum of the angles of a triangle is at most $180^\circ$. A ...
2
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1answer
115 views

What is the equation describing a three dimensional, 14 point Star?

I need to model a 14 point star. This is a three dimensional surface where there is a point at each of the eight corners of a cube and each of the six sides. The object is uniform (i.e. planar ...
6
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3answers
147 views

Evaluating $\lim_{n \to \infty}\frac{n}{2}\sqrt{2-2\cos\left(\frac{360^\circ}{n}\right)}$

I was thinking about different ways of finding $\pi$ and stumbled upon what I'm sure is a very old method: dividing a circle of radius $r$ up into $n$ isosceles triangles each with radial side length ...
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2answers
53 views

Volume of Tetrahedron

$ABCD$ is a regular tetrahedron of volume $1$. Maria glues regular tetrahedra $A'BCD$, $AB'CD$, $ABC'D$, and $ABCD'$ to the faces of $ABCD$. What is the volume of the tetrahedron $A'B'C'D'$?
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0answers
6 views

Finding the 3D coordinates of an object with cameras

I'm working on a project where I need to find the coordinates of an object. I used this paper for now http://dsc.ijs.si/files/papers/S101%20Mrovlje.pdf . It describes how with two cameras facing the ...
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1answer
15 views

Find the sum of the radii of inscribed and circumscribed circles for an n-sided in terms of cot

The sum of the radii of inscribed and circumscribed circles for an n-sided regular polygon of side 'a', is (a) $ a.cot(\frac{\pi}{n}) $ (b) $ \frac{a}{2}cot(\frac{\pi}{2n}) $ (c) $ ...
2
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3answers
60 views

Drawing circumference issue

I'm a developer, and I'm developing an app on Google Maps. At the moment, I'm trying to draw a circle on the map. For getting all the points I need, I'm using the following formula: \begin{equation} ...
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0answers
20 views

Sphericity of the 'Dual' of a spherical simplicial complex?

Suppose I have a simplicial complex $\Delta$ that is PL-homeomorphic to an $n$-sphere. Now I look at the complex $\Gamma$ that has $0$-simplices corresponding to the facets of $\Delta$ and in general ...
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3answers
40 views

Calculate pentagon area based on lengths of all its sides

Sorry for this question. I guessed there is an online calculator to calculate the area of the pentagon if we know lengths of all its five sides. So, here are the lengths of sides of pentagon ABCDE: ...
3
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2answers
89 views

Integrate area of the shadow?

Today I found an interesting article here. It computes (approximately) area of the shadow. I was wondering what is exact value of the area. My first thought was to use integrals but it doesn't seem ...
0
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1answer
53 views

$3d$ coordinate transformation with given the directions of the new coordinate

Given the original $3d$ coordinate $XYZ$. Now I got three vectors $ix, iy, iz$ in $XYZ$ each has the same direction of $x,y,z$ axis respectively of the new coordinate after transformation and the ...
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1answer
32 views

How prove that a one of sides of the hexagon has a length greater than 1?

In convex hexagon three main diagonals have a length of $> 2$. How prove that a one of sides of the hexagon has a length greater than 1?
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3answers
31 views

How do I verify that a line is parallel to a plane?

If the line $r$ has direction vector $(0,2,0)$, how can I verify if it is parallel to the following plane $\pi : x+y+z-2=0$ with orthogonal direction vector $(1,1,1)$?
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4answers
354 views

Math Competition Problem- Geometry/Calculus

So I tried the good old Calculus 1 approach and turned this into an optimization problem. The equations got REALLY hairy, but it was okay since this was the graphing calculator section of the exam. ...
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3answers
75 views

General Solution of $\sin(mx)+\sin(nx)=0$

Problem: Find the general solution of $$\sin(mx)+\sin(nx)=0$$ My attempt: $$$$ $$\sin(mx)=-\sin(nx)$$ $$=\cos\left(\dfrac{\pi}{2}-mx\right)=\cos\left(\dfrac{\pi}{2}+nx\right)$$ Using ...
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0answers
38 views

geometry of the sphere

I wish to understand the geometry of the sphere so that I can work on it for PDE problem. Could anyone suggest some good references for this (notes/books etc)? thanks
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0answers
20 views

Formal definition of mesh.

For my master thesis, I need to have a formal definition of a mesh, in a mathemathical point of view. So "geometric model described by vertices, edges and faces, ..." is not enough. And I can't seem ...
0
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1answer
25 views

Two circles covering the sides of a triangle

I would like to prove or find a counterexample for the following theorem: For any $\triangle ABC, \odot P_1, \odot P_2$ such that the three lines $AB, AC, BC$ are each contained in the union of the ...
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0answers
36 views

Provide solution to geometrical problem [closed]

Given points belonging to great circles (within an error: ±0.5⁰show): A- Which belong to the same circle? B- Which circles intersect? C- Prove the points of intersection are unique. With the data ...
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0answers
40 views

Countably Infinitely Many Points in a Euclidean Space

Do there exist $d\in\mathbb{N}$ such that there are pairwise distinct points $x_1$, $y_1$, $x_2$, $y_2$, $\ldots$ in $\mathbb{R}^d$ such that (i) $\left\|x_i-y_i\right\|_2 >1$ for ...
0
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2answers
35 views

Co-ordinate geometry involving straight lines $7x-y-32 = 0$ and $3y-2x+1=0$.

Let $P$ be the point of intersection of the lines $7x-y-32=0$ and $3y-2x+1=0$. Lines are drawn through $P$ making intercepts of equal magnitude on the co-ordinate axes. Find the equation of these ...
1
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1answer
37 views

Solve triangle point given base, point height and difference of sides

I have the intuition that one should be able to calculate the position of the circle in the image below (or the equivalent, solve a and b). We have the following information: h and d is known as well ...
0
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2answers
72 views

BMO1 2003/04 Question 2 - Geometry Prolem

$ABCD$ is a rectangle, $P$ is the midpoint of $AB$, and $Q$ is the point on $PD$ such that $CQ$ is perpendicular to $PD$. Prove that the triangle $BQC$ is isosceles. Clearly, we need to prove that ...
2
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1answer
15 views

Perspective projection of a sphere on a plane

I know the perspective projection of a sphere on a plane is an ellipse. How would I find the parametric equation for this ellipse? Say I have a camera at $(0, 0, z_2)$, a plane at $z=z_1$, and a ...
0
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1answer
32 views

Alternative proof of lateral surface area of a conical frustum

I am trying to come up with an alternative proof of the lateral surface area of a conical frustum with parallel bases by making use of the linear increase in perimeter $P$ of the base with respect to ...
0
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0answers
16 views

Maximisation of the distance of particles in a periodic box

Consider $N$ particles in a box of ratio $R=L_x/L_y$, where $L_x$ and $L_y$ are the two sides of the box. The box has periodic boundary conditions. Consider now a state which maximises the distance ...
0
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1answer
35 views

Constructing a parallelogram according to the given condition

The question #To prove two angles are equal when some angles are supplementary in a parallelogram has been solved. In the process of solving it, I found it is not that easy to draw the corresponding ...
3
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1answer
65 views

Find the shortest distance from the triangle with with vertices in $(1,1,0),(3,3,1),(6,1,0)$ to the point $(9,5,0)$.

I have to find the shortest distance from the triangle with with vertices in $(1,1,0),(3,3,1),(6,1,0)$ to the point $(9,5,0)$. I cannot figure out how to do this. There are three possible cases: ...
2
votes
2answers
51 views

Longest pipe that fits around a corner. [duplicate]

While studying, I came upon the problem "Two corridors of widths $a$ and $b$ intersect at right angle. What is the length of the longest pipe that can be carried across the two corridors, touching the ...
1
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3answers
137 views

What is the area in which the two goats can eat grass, if they choose not to eat in the common approachable area?

Two goats are tied with a rope of length 40m outside of a rectangular shed of dimensions 50m X 30m. The goats are tied to different corners which lie on the opposite ends of a diagonal of the shed. ...
4
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2answers
40 views

Area of a polygon inscribed into an ellipse

I have recently found a paper describing that the percentage area error of a polygon inscribed within a circle can be calculated using the following formula. The output of the algorithm is a set ...
2
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1answer
34 views

How can I prove that modul of this vector bigger than radius of a circle?

Let be $A$, $B$, $C$ be three points on the semicircle with diameter $PQ=2$ ($O$ is center) . Prove that modul of vector $\overrightarrow{OA}+\overrightarrow{OB}+\overrightarrow{OC}$ bigger than 1. I ...
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2answers
21 views

equation of projection onto hyperplane

Let $P$ be a projection mapping onto the hyperplane trough the origin which is normal to $v$. How do you show that $Px=x-\dfrac{vv^T}{v^Tv}x$ Any intuition?
2
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0answers
28 views

Proof-validation of centroid's existence

So, a friend of mine came up with this unorthodox proof of the centroid's existence so I figured I could share it here so that someone can confirm that it's a fine one. I think it is correct, but I ...
5
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1answer
196 views

IMO 2015 #1: “Balanced” and “Centre-Free” sets of points in the plane [closed]

International Mathematical Olympiad 2015, Problem 1: We say that a finite set $S$ of points in the plane is $\color{\red}{\text{balanced}}$ if, for any two different points $A$ and $B$ in $S$, ...
2
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4answers
120 views

how to prove that the circle $(x-a)^2+(y-b)^2=a^2+b^2$ is passing through point $(0,0)$

How can one prove that the circle $(x-a)^2+(y-b)^2=a^2+b^2$ is passing through point $(0,0)$? I know that if i put: $x=y=0$, i will get: $(0-a)^2+(0-b)=a^2+b^2=a^2+b^2$ But that's not a proof but ...
2
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1answer
57 views

P. Winkler's puzzle “Inscribing a Lake in a Square”

This is a puzzle from P. Winkler: "Show that, given any closed curve in the plane, there is a square containing the curve, all four sides of which touch the curve." I was NOT able to solve it quickly ...
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0answers
33 views

Special Non-linear recurrence

Problem I have a non-linear recurrence relation given by $$ a_n = a_{n-1}+a_{n-2}+a_{n-3} - \sqrt{a_{n-1}.a_{n-2}+a_{n-2}.a_{n-3}+a_{n-3}.a_{n-1}} $$ Given $ a_1, a_2 $ and $ a_3 $,I have to find ...
2
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1answer
101 views

Find angles of two intersected planes.

I'm really bad at math so I'll try to explain as best as I can. Here's a visual representation of what I need to do. Basically it's a pop-up book. There is a plane which can be folded on the blue ...
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4answers
105 views

Median of triangle

I know that a median of a triangle is a line joining one of the vertices to the mid-point of the opposite side. For example, in a triangle OAB, O is the origin, $A$ is the point $(0,6)$ and $B$ is ...
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2answers
38 views

Maximum sum of cathetii (shorter arms of triangle ) with hypotenuse length =1

A rectangular triangle has its hypotenuse with length = 1. Prove that the sum of the shorter sides will be maximum when the triangle is isosceles. I have no ideas how to prove it.
5
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1answer
89 views

Could Euclid have proven that multiplication of real numbers distributes over addition?

In Euclid's day, the modern notion of real number did not exist; Euclid did not believe that the length of a line segment was a quantity measurable by number. But he did think it made sense to talk ...
3
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1answer
22 views

Significance of homothety mapping incircle to circumcircle

Are there any special properties of the homothety mapping the incircle of a triangle to the circumcircle? For example are the centers of this homothety triangle centers?
0
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1answer
28 views

How to construct center of homothety for two circles which overlap

In general any two circles have two centers of homothety. They have only one center when the circles have the same radius or when the circles have the same center. Given two circles of different ...
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1answer
117 views

Could Euclid have proven that real number multiplication is commutative?

In Euclid's day, the modern notion of real number did not exist; Euclid did not believe that the length of a line segment was a quantity measurable by number. But he did think it made sense to talk ...
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0answers
44 views

Overlap between two angles

Imagine that we have $4$ line segments in a plane. The starting point of all segments is the same and end point of them could be any arbitrary value . We call the angle between the segments $1,2 ...
6
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2answers
76 views

Why is pi/2 the angle associated with orthogonality in euclidean geometry?

This is sort of a weird question, but I am trying to understand why 90 degrees or pi/2 radians is the angle that corresponds to orthogonality in 3-d (or really any dimension of) Euclidean space. Said ...